Integrating a function that includes a series expansion

Hello everyone. First of all, greetings, this is my first time here. I have just begun a chemistry master's program in Germany.

For my question, I am not looking for the solution for free, but more assistance on where I should even begin. The problem and the final equation are both given in the lecture notes; unfortunately, the gap between of how to get to the solution was not provided.

Without further ado:
problem

I could really use a tip in the right direction. I have been refreshing my mind of geometric series, power series, and Taylor series. My first inclination is to try to rewrite the series w(k) as the function it represents. I stumble here, because it does not quite resemble a Taylor series since 1/n! is not present, and if I attempt to correlate it with a geometric or power series of the form a+ax+ax^2... (origin at zero for simplicity), I cannot match the derivative coefficient in w(k) to the coefficient "a" from a power series.

In other words, I don't know how to plug in w(k) into the given equation in a way that I can then solve the integral. I've tried working backwards from the final equation. I can follow it mostly, except that I don't know where the sin function comes from. The Taylor series of a sin function has alternating positive and negative terms, which is not a feature of w(k).

It's driving me crazy. I'd appreciate any links, starting points, words of advice that can be given. Thank you very much!

Re: Integrating a function that includes a series expansion

I AM SO CLOSE. Thanks a lot for your response, Chiro! I kept trying to plug in the series expansion to infinity, and I had no idea how to solve the subsequent integral. But if I just take the first two terms, my answer comes out very close to what was shown in the handout to be the answer.

Original problem:

I am stuck, though, and maybe I am missing something very obvious. I have reached the final equation, only the denominator term is reversed ie.

instead of: w't-x
I have: x-w't

The same switch has occurred in my sin function as well.

I backtracked my work, and the denominator term comes from the chain rule when I integrate the original function. My original function is simply the first two terms of the Taylor expansion substituted in for w(k).

In other words, it looks like this:

I think that this is where the error comes from, namely the denominator is already reversed in sign.

The rest of my work is as follows

And the final answer is: (the last equation from the above image was reproduced as the first equation of the following image)

I am not seeing where my mistake is, but maybe I have been looking at this problem too long now. Typical of me in such a situation, I am, of course, beginning to suspect the handout is wrong If anyone has time or interest to check the work, could you tell me if you see where I might be messing up? Thank you!

EDIT: Ok, well, someone has advised me that the sin function is an odd function, so I can pull out a negative from it and reverse the sign of the denominator as well. This leads to the correct answer!